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SandBoxWirtinger

Edit detail for SandBoxWirtinger revision 20 of 21

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Editor: Bill Page Time: 2015/03/16 21:55:35 GMT+0
Note:

added:

- "Non-analytic functions of a complex variable":http://www.ams.org/journals/bull/1933-39-02/S0002-9904-1933-05562-3/S0002-9904-1933-05562-3.pdf
  or "alternate":http://projecteuclid.org/download/pdf_1/euclid.bams/1183496499
  by E. R. Hedrick

- "Pseudo-Conformal Geometry of Polygenic Functions of Several Complex Variables":http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063264/pdf/pnas01560-0079.pdf
  by Edward Kasner and John De Cicco

- "A Complete Characterization of the Derivative of a Polygenic Function":http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076732/pdf/pnas01767-0024.pdf
  by Edward Kasner

- "Polygenic Functions Of Hypercomplex Variables":http://www.ams.org/journals/bull/1932-38-02/S0002-9904-1932-05329-0/S0002-9904-1932-05329-0.pdf
  by P.W.Ketchum and Ted Martin

Wirtinger Derivatives or Wirtinger Calculus (also called $\mathbb{CR}$ -Calculus)

Reference

http://en.wikipedia.org/wiki/Wirtinger_derivatives
The Complex Gradient Operator and the CR-Calculus by Ken Kreutz-Delgado
Appendix A: Wirtinger Calculus by Robert F. H. Fischer
Wirtinger derivatives, Beltrami equation & ellipse fields by Thomas Schramm
Non-analytic functions of a complex variable or alternate by E. R. Hedrick
Pseudo-Conformal Geometry of Polygenic Functions of Several Complex Variables by Edward Kasner and John De Cicco
A Complete Characterization of the Derivative of a Polygenic Function by Edward Kasner
Polygenic Functions Of Hypercomplex Variables by P.W.Ketchum and Ted Martin

fricas

)lib FSPECX

   FunctionalSpecialFunction is now explicitly exposed in frame initial

   FunctionalSpecialFunction will be automatically loaded when needed 
      from /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX

fricas

diff(ex,z) == eval(D(eval(ex,[z],[%conjugate]),%conjugate),[%conjugate],[z])

Type: Void

Tests

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test(diff(z,z)=1)

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Compiling function diff with type (Variable(z),Variable(z)) -> 
      Polynomial(Integer)

$\label{eq1} \mbox{\rm true}$

(1)

Type: Boolean

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test(diff(conjugate z,conjugate z)=1)

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Compiling function diff with type (Expression(Integer),Expression(
      Integer)) -> Expression(Integer)

$\label{eq2} \mbox{\rm true}$

(2)

Type: Boolean

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test(diff(z,conjugate z)=0)

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Compiling function diff with type (Variable(z),Expression(Integer))
       -> Expression(Integer)

$\label{eq3} \mbox{\rm true}$

(3)

Type: Boolean

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test(diff(conjugate z,z)=0)

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Compiling function diff with type (Expression(Integer),Variable(z))
       -> Expression(Integer)

$\label{eq4} \mbox{\rm true}$

(4)

Type: Boolean

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diff(sin(x),x)

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Compiling function diff with type (Expression(Integer),Variable(x))
       -> Expression(Integer)

$\label{eq5}\cos \left({x}\right)$

(5)

Type: Expression(Integer)

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diff(sin(x),conjugate x)

$\label{eq6}0$

(6)

Type: Expression(Integer)

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diff(sqrt(x^2),x)

$\label{eq7}x \over{\sqrt{{x}^{2}}}$

(7)

Type: Expression(Integer)

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diff(sqrt(x^2),conjugate x)

$\label{eq8}0$

(8)

Type: Expression(Integer)

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diff(z*conjugate(z),z)

$\label{eq9}\overline z$

(9)

Type: Expression(Integer)

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diff(z*conjugate(z),conjugate z)

$\label{eq10}z$

(10)

Type: Expression(Integer)

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diff(sqrt(x*conjugate(x)),x)

$\label{eq11}{\overline x}\over{2 \ {\sqrt{x \ {\overline x}}}}$

(11)

Type: Expression(Integer)

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diff(%,x)

$\label{eq12}-{{\overline x}\over{4 \ x \ {\sqrt{x \ {\overline x}}}}}$

(12)

Type: Expression(Integer)

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diff(%,x)

$\label{eq13}{3 \ {\overline x}}\over{8 \ {{x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(13)

Type: Expression(Integer)

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diff(diff(diff(abs(x),x),x),x)

$\label{eq14}{3 \ {{\overline x}^{3}}}\over{8 \ {{abs \left({x}\right)}^{5}}}$

(14)

Type: Expression(Integer)

fricas

_rule(abs(x),sqrt(x*conjugate(x))) %

$\label{eq15}{3 \ {\overline x}}\over{8 \ {{x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(15)

Type: Expression(Integer)

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diff(sqrt(x*conjugate(x)),conjugate x)

$\label{eq16}x \over{2 \ {\sqrt{x \ {\overline x}}}}$

(16)

Type: Expression(Integer)

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conjugate(diff(conjugate(sqrt(x*conjugate(x))),x))

$\label{eq17}{\sqrt{x \ {\overline x}}}\over{2 \ {\overline x}}$

(17)

Type: Expression(Integer)

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diff(%,conjugate x)

$\label{eq18}-{x \over{4 \ {\overline x}\ {\sqrt{x \ {\overline x}}}}}$

(18)

Type: Expression(Integer)

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diff(%,conjugate x)

$\label{eq19}{3 \ x}\over{8 \ {{\overline x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(19)

Type: Expression(Integer)

Non-holomorphic functions

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real(x) == (x+conjugate(x))/2

Type: Void

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test(real real x = real x)

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Compiling function real with type Variable(x) -> Expression(Integer)

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Compiling function real with type Expression(Integer) -> Expression(
      Integer)

$\label{eq20} \mbox{\rm true}$

(20)

Type: Boolean

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imag(x) == -%i*(x-conjugate(x))/2

Type: Void

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test(imag imag x = 0)

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Compiling function imag with type Variable(x) -> Expression(Complex(
      Integer))

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Compiling function imag with type Expression(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq21} \mbox{\rm true}$

(21)

Type: Boolean

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real(a+%i*b)

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Compiling function real with type Polynomial(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq22}{-{i \ {\overline b}}+{\overline a}+{i \ b}+ a}\over 2$

(22)

Type: Expression(Complex(Integer))

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eval(%,[conjugate(a)=a,conjugate(b)=b])

$\label{eq23}a$

(23)

Type: Expression(Complex(Integer))

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test(real(real(a) + %i*real(b)) = real(a))

fricas

Compiling function real with type Variable(a) -> Expression(Integer)

fricas

Compiling function real with type Variable(b) -> Expression(Integer)

fricas

Compiling function real with type Expression(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq24} \mbox{\rm true}$

(24)

Type: Boolean

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imag(a+%i*b)

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Compiling function imag with type Polynomial(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq25}{{\overline b}+{i \ {\overline a}}+ b -{i \ a}}\over 2$

(25)

Type: Expression(Complex(Integer))

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eval(%,[conjugate(a)=a,conjugate(b)=b])

$\label{eq26}b$

(26)

Type: Expression(Complex(Integer))

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diff(real(x),x)

$\label{eq27}1 \over 2$

(27)

Type: Expression(Integer)

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diff(real(x),conjugate x)

$\label{eq28}1 \over 2$

(28)

Type: Expression(Integer)

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diff(imag(x),x)

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Compiling function diff with type (Expression(Complex(Integer)),
      Variable(x)) -> Expression(Complex(Integer))

$\label{eq29}-{i \over 2}$

(29)

Type: Expression(Complex(Integer))

fricas

diff(imag(x),conjugate x)

fricas

Compiling function diff with type (Expression(Complex(Integer)),
      Expression(Integer)) -> Expression(Complex(Integer))

$\label{eq30}i \over 2$

(30)

Type: Expression(Complex(Integer))

fricas

diff(abs(x)^2,x)

$\label{eq31}\overline x$

(31)

Type: Expression(Integer)

Functions are assumed holomorphic

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f:=operator 'f

$\label{eq32}f$

(32)

Type: BasicOperator?

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diff(c*f(x),x)

$\label{eq33}c \ {{f_{\ }^{,}}\left({x}\right)}$

(33)

Type: Expression(Integer)

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diff(c*f(x),conjugate x)

$\label{eq34}0$

(34)

Type: Expression(Integer)

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diff(f(x,conjugate x),x)

$\label{eq35}{f_{, 1}}\left({x , \:{\overline x}}\right)$

(35)

Type: Expression(Integer)

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diff(f(x,conjugate x),conjugate x)

$\label{eq36}{f_{, 2}}\left({x , \:{\overline x}}\right)$

(36)

Type: Expression(Integer)

fricas

test(diff(conjugate f(z,conjugate z),conjugate z) = conjugate diff(f(z,conjugate z),z))

$\label{eq37} \mbox{\rm false}$

(37)

Type: Boolean

fricas

test(diff(conjugate f(z,conjugate z),z) = conjugate diff(f(z,conjugate z),conjugate z))

$\label{eq38} \mbox{\rm false}$

(38)

Type: Boolean

Examples

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t1:=real(z)^2+%i*imag(z)^3

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Compiling function real with type Variable(z) -> Expression(Integer)

fricas

Compiling function imag with type Variable(z) -> Expression(Complex(
      Integer))

$\label{eq39}{{{\overline z}^{3}}+{{\left(-{3 \ z}+ 2 \right)}\ {{\overline z}^{2}}}+{{\left({3 \ {{z}^{2}}}+{4 \ z}\right)}\ {\overline z}}-{{z}^{3}}+{2 \ {{z}^{2}}}}\over 8$

(39)

Type: Expression(Complex(Integer))

fricas

test(diff(t1,z) = 1/2*z+1/2*conjugate(z)-3/8*(z-conjugate(z))^2)

fricas

Compiling function diff with type (Expression(Complex(Integer)),
      Variable(z)) -> Expression(Complex(Integer))

$\label{eq40} \mbox{\rm true}$

(40)

Type: Boolean

fricas

(diff(t1,conjugate z) = 1/2*z+1/2*conjugate(z)-3/8*(z-conjugate(z))^2)

$\label{eq41}\begin{array}{@{}l} \displaystyle {{{3 \ {{\overline z}^{2}}}+{{\left(-{6 \ z}+ 4 \right)}\ {\overline z}}+{3 \ {{z}^{2}}}+{4 \ z}}\over 8}= \ \ \displaystyle {{-{3 \ {{\overline z}^{2}}}+{{\left({6 \ z}+ 4 \right)}\ {\overline z}}-{3 \ {{z}^{2}}}+{4 \ z}}\over 8}$

(41)

Type: Equation(Expression(Complex(Integer)))

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test(diff(t1,conjugate z) = 1/2*z+1/2*conjugate(z)-3/8*(z-conjugate(z))^2)

$\label{eq42} \mbox{\rm false}$

(42)

Type: Boolean

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t2:=2*log(sqrt(z*conjugate(z)))

$\label{eq43}2 \ {\log \left({\sqrt{z \ {\overline z}}}\right)}$

(43)

Type: Expression(Integer)

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diff(t2,z)

$\label{eq44}1 \over z$

(44)

Type: Expression(Integer)

fricas

diff(t2,conjugate z)

$\label{eq45}1 \over{\overline z}$

(45)

Type: Expression(Integer)

fricas

diff(sqrt(x*conjugate(x))/x,x)+diff(sqrt(x*conjugate(x))/x,conjugate x)

$\label{eq46}{-{\overline x}+ x}\over{2 \ x \ {\sqrt{x \ {\overline x}}}}$

(46)

Type: Expression(Integer)

fricas

eval(%,conjugate(x)=x)

$\label{eq47}0$

(47)

Type: Expression(Integer)